{
  "abstract": "We minimize $\\mathbb E[h(Y-X)]$ over martingale couplings of $\\text{Unif}[-1,1]$ and $\\text{Unif}[-2,2]$ satisfying $Y\\geq X-k$, where $h\\in C^1([-3,3])$ has convex derivative. Feasibility holds exactly for $k\\geq1$. For each such $k$, we construct a coupling that minimizes all costs in this class. For $1<k<3$, its support consists of two graphs, with $D(x)=x-k$ on $[k-2,1]$. The maps admit an explicit parametrization for $2\\leq k<3$ and are determined by scalar equations with unique admissible roots for $1<k<2$. We prove optimality by a dual inequality, using an analytic estimate in the latter range. At $k=1$ the optimizer is $Y=X\\pm1$ with equal probabilities; for $k\\geq3$ it is the ordinary left-curtain coupling. In the unconstrained problem, left-monotonicity identifies the left-curtain coupling, which is optimal for this cost class. Under the constraint, a discrete example shows that the corresponding support condition, even together with every two-source comparison, does not imply optimality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29997",
  "author_total": 2,
  "authors": [
    {
      "name": "Erhan Bayraktar"
    },
    {
      "name": "Xin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/341pdky0r5r95.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29997",
  "fulltext_url": null,
  "id": "341pdky0r5r95",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29997",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214359729",
  "pdf_url": "https://arxiv.org/pdf/2609.29997",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/341pdky0r5r95.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29997",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29997"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A one-sided constrained martingale transport between two uniform laws",
  "title_text": "A one-sided constrained martingale transport between two uniform laws",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/341pdky0r5r95",
  "venue": "arXiv",
  "versions": []
}